Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch

Monday, July 29, 2013

Certifying primality in a CONSTANT number of operations

For this post I will only count the operations PLUS, MINUS, MULT. They may be done on rather large numbers.

Recall that from the work coming out of Hilberts 10th problem we know the following: For every c.e. set (used to be called r.e., some people still do) there is a polynomial f in 13 or less variables (we'll assume 13) with coefficients in the integers such that

x in A iff (∃ a1,...,a13))[f(x,a1,...,a13)=0]

In an article about Hilbert's 10th problem written in 1974 by Davis-Matiyasevich-Robinson they note that by this result there is a FINITE number M such that, for ALL primes p, there is a certification that p is prime that uses at most M operations: given p a prime let a1,...,a13 be such that f(p,a1,...,a13)=0. The certification that p is prime is just the evaluation of that polynomial and seeing that its 0.

Is this still the only proof that one can certify primality in a CONSTANT number of operations?

Primes is irrelevant to all of this--- any c.e. set would work. (The result for c.e. sets may qualify as a theorem that is less interesting because its more interesting.) But for primes I am wondering if there is another way to do this- perhaps using number theory, perhaps with a smaller value of M. For the explicit poly for primes, due to Jones, see here.

There may be something here to wonder about for complexity theory. Most people didn't believe that the r.e. sets are diophantine since that would have so many unlikely consequences, as e.g. that the primes are diophantine, none of which were believed to hold, and were far from being proved to hold. In fact, Matiyasevich's result came rather unexpected and was of the P=NP-type of resolving Hilbert's 10-th problem.

My experience is that excluding GCT almost no-one works on a direct proof or disproof. If you add research towards proving the hypothesis of conditional result that would then give (usually) P <> NP , you'll get probably about 100 researchers or so.

Among logicians, the Matiyasevich result was not a surprise. There were several people working on it at the time, and the existing partial results of Julia Robinson were tantalizingly close. Martin Davis had even gone so far as to predict that some young Russian was going to complete the existing work (his point being that the end was in sight, but horribly tedious).